What is the integral of cos(xt) ?

The integral of cos(xt) is 1/x sin(xt)+C

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integral of cos(xt)

\int cos (x t) d t

\frac{1}{x} sin (x t) + C

Solution steps

\int cos (x t) d t

Apply u-substitution

: \int \frac{cos ( u )}{x} d u

= \int \frac{cos ( u )}{x} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{x} \cdot \int cos (u) d u

Use the common integral:

\int cos (u) d u = sin (u)

= \frac{1}{x} sin (u)

Substitute back

u = x t

= \frac{1}{x} sin (x t)

Add a constant to the solution

= \frac{1}{x} sin (x t) + C

\int \frac{1}{8} d x

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